## Mixing Length Eddy Viscosity and the w Spectrum

In Section 3.3 we introduced a length scale of the energy-containing eddies, X, and used it in the expression for shear production given by (3.5). Discussion of IOBL turbulence scales is the subject of Chapter 5, but here we anticipate those results by identifying X with the wave number at the peak in the area-preserving w spectrum, kmax, and introduce the concept of eddy viscosity in a turbulent flow. Eddy viscosity provides a conceptual method for closing the turbulence problem at "first-order" by relating fluxes of momentum and scalar properties in a shear flow to their respective gradients. By analogy with kinematic viscosity which depends on the products of velocity (internal energy) and mean free path of the molecules in the fluid, eddy viscosity is commonly represented by the product of a turbulent velocity scale and a length scale over which the dominant eddies in a flow are effective at diffusing momentum (this is different from the actual scale of the eddy motions):

In the simplest form of turbulent shear flow near a boundary where buoyancy and rotation are unimportant, the velocity profile is logarithmic with distance from the boundary, and the pertinent scale velocity is u*0, the square root of kinematic boundary stress. From this it follows immediately (Section 4.1) that X = Kz where K is von Karman's constant. If e is known in this type of flow (say from measuring the spectrum), the magnitude of the turbulent stress is simply T = (Kze)2/3. The problem for the IOBL, however, is that the linear dependence of X on z is limited to at most a few meters from the boundary, and determining the scale of turbulence in the outer part of the boundary layer becomes a central issue in understanding turbulent transfer there. It appears that the inverse of the wave number at the peak in the w spectrum (but not the u and v spectra) provides a consistent estimate of X, hence eddy viscosity. Measuring this at discrete levels through the entire IOBL then provides an important observational constraint on models that purport to simulate turbulent exchanges in the PBL.

Application of this concept to the IOBL dates from the 1972 AIDJEX Pilot Study data (McPhee and Smith 1976). We found that peaks in the area-preserving spectral density of vertical velocity variance increased with depth for the first couple of turbulence clusters (to about 4 m from the ice) but that for greater depths (8-26 m), the peaks occurred at roughly the same wave number. We were also aware that Busch and Panofsky (1968) had shown that velocity variance spectra measured in the atmospheric boundary layer had the following characteristics: (i) wavelengths at the maximum in the logarithmic (area-preserving) spectra of vertical velocity increased linearly with height up to about 50 m, and more slowly beyond; (ii) the dimension-less frequency, fm = nz/V where n is frequency at the spectral maximum and V is mean wind speed, scaled with Monin-Obukhov similarity in the surface layer (discussed in Chapter 4); (iii) there was a relatively uniform shape to the normalized w spectra, when the abscissa values were scaled by fm and ordinate values by u20; and (iv) the longitudinal (u) spectra did not show similarly predictable behavior. For the neutrally stratified surface layer, this means that the wave number at the maximum in the area-preserving w spectrum (kmax = fm/z) is inversely proportional to z hence to Xs¡ = Kz. The fact that this dependence weakened for heights greater than about 50 m led Busch and Panofsky to speculate that the connection between X and kmax might persist beyond the surface layer.

We reasoned (McPhee and Smith 1976) that if the increase in X with distance from the boundary reached some limit comparable to K times the surface layer thickness (a few meters in the IOBL), then our observations of kmax behavior at the various levels would be consistent with X = cx/kmax, as suggested by Busch and Panofsky. Experiments since (described in Chapter 5) have corroborated this view. The proportionality constant appears to be about 0.85. The vertical line in Fig. 3.9 thus suggests that the master turbulent length scale at 20 m during the 6-h event shown was about 3.2 m. If shear production and dissipation balance, (3.5) then implies that the Reynolds stress, T = (Xe)2/3, was about 5 x 10 4m2 s , in agreement with direct measurements (see Fig. 5.3). Note that there is enough information from the w spectrum by itself to estimate the eddy viscosity at 20 m: K « e1/3X4/3, 2 — 1

which is about 0.022m2 s .

The isotropy condition indicated by the 4/3 separation between w and u spectra as shown by the ISW data (Fig. 3.9) is not always present in the IOBL measurements, particularly at levels closer to the surface. During the SHEBA project, for example, average normalized w spectra were remarkably similar at four levels ranging from 4 to 16 m from the ice undersurface (see Fig. 3.6 of McPhee 2004). However, for the two upper TICs, the u spectrum was consistently more energetic in the inertial subrange (as indicated by the w spectrum). We attributed this to a lack of horizontal homogeneity in the underice surface, as TKE advected from a prominent pressure ridge keel, often about 100 m "upstream" from the turbulence mast, spread vertically. In this case, we found that the gradient of TKE flux played a significant role in the TKE equation, and developed an alternative (but closely related) method for estimating the magnitude of stress from the w spectrum by considering the production rather than dissipation of TKE, resulting in a simple relation

0.025 0.02

0.01

0.005

0.025 0.02

0.01

 covar estimate spectral estimate ■ ------ 258 259 260 261 260 261 Day of 1998 262 263 Fig. 3.10 Friction velocity as measured by direct covariance (^u* = (^{u'w1}2 + (v'w'}2^ 260 261 Day of 1998 262 263 Fig. 3.10 Friction velocity as measured by direct covariance (^u* = (^{u'w1}2 + (v'w'}2^ eraged in 3-h blocks (black *) and derived from the w spectrum as described in the text (grey squares) at two levels near the end of the SHEBA project. Dashed (covariance) and dot-dashed (spectra) horizons show mean values (Adapted from McPhee 2004. With permission of the American Meteorological Society) where 0 = kSw(k), evaluated at wave number k = Y*kmax where Y* is a wave number in the —2/3 spectral range of the log-log w spectrum, normalized by the wave number at the maximum. We found the proportionality constant to be Cy = 0.48. For SHEBA Y* = 2.5(logy* = 0.4) was consistently in the —2/3 range. Details of the derivation are presented in McPhee (2004). A comparison of friction velocity calculated via the spectral method (3.12) with direct covariance estimates for a time late in the SHEBA project is shown in Fig. 3.10.